The starting point for our work is the framework in [1]}
which views the task of decoding \(\tilde{y}\)  with
\(\Delta (\mathcal {C}^{\prime },\tilde{y}) < (1-\varepsilon )/4 - \delta \)  (where the distance of
\(\mathcal {C}^{\prime }\)  is \((1-\varepsilon )/2\) ) as an instance of the \(\textsf {MAX t-XOR} \)  problem
(see fig:approxuniquedec).
The goal is to find
\(\operatornamewithlimits{\arg \!\min }_{z \in \mathcal {C}}\Delta \left(\operatorname{dsum}_{W(t)}(z), \tilde{y}\right),\) 
